Abstract

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field of -complex numbers and the concept of -metric. Also, we give the definitions and the basic important properties of -boundedness and -continuity. Later, we define the space of -continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between and the set of -differentiable functions.

1. Preliminaries, Background and Notations

As a popular non-Newtonian calculus, multiplicative calculus was studied by Stanley [1] in a brief overview. Bashirov et al. [2] have recently emphasized on the multiplicative calculus and gave the results with applications corresponding to the well-known properties of derivative and integral in the classical calculus. Recently, in [3], the multiplicative calculus has extended to the complex valued functions and interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. Bashirov and Riza [4] have studied on the multiplicative differentiation for complex valued functions and established the multiplicative Cauchy-Riemann conditions. Bashirov et al. [5] have investigated various problems from different fields which can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus. Quite recently, Çakmak and Başar [6] have showed that non-Newtonian real numbers form a field with the binary operations addition and multiplication. Further, the non-Newtonian exponent, surd, and absolute value are defined and some of their properties are given. They also proved that the spaces of all bounded, convergent, null and absolutely p-summable sequences of the non-Newtonian real numbers are the complete metric spaces. Quite recently, Tekin and Başar have [7] proved that the corresponding classical sequence spaces are Banach spaces over the non-Newtonian complex field. Quite recently, Çakir [8] has defined the sets and of geometric complex valued bounded and continuous functions and showed that and form a vector space with respect to the addition and scalar multiplication in the sense of multiplicative calculus and are complete metric spaces, where denotes the compact subset of the complex plane . Quite recently, Uzer [9] has investigated the waves near the edge of a conducting half plane. Firstly, the series is converted into contour integrals in a complex plane and then some contour deformations are made. After that, the resultant integrals are converted back into the series forms, which are seen to be rapidly convergent near the reflection/shadow boundaries of the conducting half plane. In the second part, a multiplicative calculus is employed for evaluating the relevant integrals, approximately. By the way, he derives a simple expression, which can be used whenever the series derived in the first part of the paper is not rapidly convergent.

Non-Newtonian calculus is an alternative to the usual calculus of Newton and Leibniz. It provides differentiation and integration tools based on non-Newtonian operations instead of classical operations. Every property in classical calculus has an analogue in non-Newtonian calculus. Generally speaking, non-Newtonian calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus. In some cases, for example, for wage-rate- (in dollars, euro, etc.) related problems, the use of bigeometric calculus which is a kind of non-Newtonian calculus is advocated instead of a traditional Newtonian one.

Throughout this paper, non-Newtonian calculus is denoted by (NC), and classical calculus is denoted by (CC). Also for short we use -continuity for non-Newtonian continuity. A generator is a one-to-one function whose domain is and whose range is a subset of . Each generator generates exactly one type of arithmetic, and conversely each type of arithmetic is generated by exactly one generator. As a generator, we choose the function from to the set that is to say that

In the special cases and , generates the classical and geometric arithmetic, respectively, where denotes the identity function whose inverse is itself. The set of non-Newtonian real numbers are defined by .

Following Bashirov et al. [2] and Uzer [3], the main purpose of this paper is to construct the space of non-Newtonian complex valued continuous functions which forms a Banach space with the norm defined on it. Finally, we give some applications to seek how (NC) can be applied to the classical Functional Analysis problems such as approximation and inner product properties.

We should know that all concepts in classical arithmetic have natural counterparts in -arithmetic. For instance, -zero and -one turn out to be and . Similarly, the -integers turn out to

Consider any generator with range . By -arithmetic, we mean the arithmetic whose domain is and the operations are defined as follows: for and any generator ,

Particularly, if we choose , the identity function, as an -generator, then and for all and therefore -arithmetic obviously turns out to the classical arithmetic. Consider
If we choose as an -generator defined by for , then and -arithmetic turns out to geometric arithmetic. Consider

Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset of . There are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent. Nevertheless, the fact that two systems are isomorphic does not exclude their separate usage. In [2], it is shown that each ordered pair of arithmetic give rise to a calculus by a sensible use of the first arithmetic or function arguments and the second arithmetic for function values.

Let and be arbitrarily selected generators and is the ordered pair of arithmetic. Table 1 may be useful for the notation used in -arithmetic and -arithmetic.

Table 1: Notation in -arithmetic and -arithmetic.

Definitions for -arithmetic are also valid for -arithmetic. For example, -convergence is defined by means of -intervals and their -interiors.

In the (NC), -arithmetic is used for arguments and -arithmetic is used for ranges; in particular, changes in arguments and ranges are measured by -differences and -differences, respectively. The operators of the (NC) are applied only to functions with arguments in and values in .

The isomorphism from -arithmetic to -arithmetic is the unique function (iota) which has the following three properties:(i)is one to one;(ii) is on and onto ;(iii)for any numbers and in ,
It turns out that for all in and that for every integer .

Since, for example, , it should be clear that any statement in -arithmetic can readily be transformed into a statement in -arithmetic.

Throughout this paper, we define the -th -exponent and the -th -root of by
and provided there exists an such that .

The -absolute value of a number in is defined as and is denoted by .

For each -nonnegative number , the symbol will be used to denote which is the unique -nonnegative number whose -square is equal to . For each number in ,
where the absolute value of is defined by

The -distance between two points and is defined by and has the symmetry property, since

Let any be given. Then, is called a positive non-Newtonian real number if , is called a non-Newtonian negative real number if , and, finally, is called an unsigned non-Newtonian real number if . By and , we denote the sets of non-Newtonian positive and negative real numbers, respectively.

In (CC), we have and for . The following lemmas show that the corresponding results also hold in non-Newtonian calculus.

Let be an infinite sequence of the elements in . Then, there is at most one element in such that every -interval with in its -interior contains all but finitely many terms of . If there is such a number , then is said to be -convergent to , which is called the -limit of . In other words,
The -limit of a function at an element in is, if it exists, the unique number in such that, for every infinite sequence of arguments of distinct from , if is -convergent to , then -converges to and is denoted by . That is,
A function is -continuous at a point in if and only if is an argument of and . When and are the identity function , the concepts of -limit and -continuity are identical with those of classical limit and classical continuity.

The -change of a function over an interval is the number . A -uniform function is a function in , is -continuous, and has the same -change over any two -interval of equal -extent. The -uniform functions are those expressible in the form , where and are constants in and is unrestricted in . By choosing and , we see that is -uniform. It is characteristic of a -uniform function that, for each -progression of arguments, the corresponding sequence of values is a -progression. The -slope of a -uniform function is its -change over any interval of -extent . For example, the -slope of the function turns out to be . In particular, the -slope of equals , and the -slope of a constant function on equals .

The -gradient of a function over is the -slope of the -uniform function containing and showed as and turns out to be

If the following -limit exists, the -derivative of at , and say that is -differentiable at ,
If it exists, is necessarily in .

The -derivative of is the function that assigns to each number in the number , if it exists.

The classical derivatives and do not necessarily coexist and are seldom equal; however, if the following exist,
then both and exist.

We denote the sets of -bounded functions, -continuous functions and -differentiable functions in the -closed interval by , , and , respectively.

The -average of a -continuous function on is denoted by and defined to be the -limit of the -convergent sequence whose th term is the -average of , where is the -fold -partition of . The -average of a -uniform function on is equal to the -average of its values at and and is equal to its value at the -average of and .

The -integral of a -continuous function on , is the following number in :

It is trivial that

Since
the -integral is a weighted -average.

Furthermore, equals to the -limit of the -convergent sequence whose th term is
where is the -fold partition of -partition of and is the common value of .

If is classically continuous function and , then the -integral is a Stieltjes integral.

Theorem 3. The -derivative and -integral are inversely related in the sense indicated by the following two statements. (i)If is -continuous on and for every , then on .(ii)If is -continuous on , then .

It is convenient to indicate the uniform relationships between the corresponding notions of the -calculus and classical calculus.

For each number , let . Let be a function from into , and set . Then, both and exist and

Furthermore, is -continuous at if and only if is classically continuous at .

If is the -gradient of over , then , where is the classical gradient of over .

If both and exist, then we have . If is -continuous on , then and

The rest of the paper is organized as follows.

In Section 2, it is shown that the set of non-Newtonian complex (complex) numbers forms a field with the binary operations addition and multiplication . Further, some basic properties and inequalities which play the basic role in -convergence and -continuity are proved. Section 3 is devoted to the space of -continuous functions of a -complex variable. We prove that is a complete metric space with the natural metric and is a Banach space with the natural norm and the space of all -bounded mappings from into is a Banach space. As an application part, in Section 4, we try to create the -inner product space specifically for (MC) and give an inclusion relation between and the set of -differentiable functions. In the final section of the paper, we note the significance of the (NC) and record some further suggestions.

2. -Complex Field and -Inequalities

In this section, following Tekin and Başar [7], we give some knowledge on the -complex field and some concerning inequalities.

Let and be arbitrarily chosen elements from corresponding arithmetic. Then, the ordered pair is called as a -point. The set of all -points is called the set -complex numbers and is denoted by , that is,
Define the binary operations addition and multiplication of -complex numbers and as follows:
where and with
Then, Tekin and Başar [7, Theorem 2.1] proved that is a field.

The -distance between any two elements and of the set is defined by
Here and after, we know that is a field and the distance between two points in is computed by the relation . Now, we will see whether this relation is metric over or not, define -norm, and try to obtain some required inequalities in the sense of non-Newtonian complex calculus.

Theorem 7 (see [7, Theorem 2.6]). is a complete metric space, where is defined by (27).

In this paper, we mainly focus on the metric on the -complex numbers because -continuity always required to use that metric relation. Therefore, we present the completeness of the set with respect to the metric.

Theorem 8. is a Banach space with the norm defined by
where and .

3. Continuous Function Space overthe Field

In this section, we construct the space of continuous functions over the field and show that this space is a complete metric space with metric such that
It would not be too hard to find out that the space of -continuous functions creates a normed space with the norm reduced from the metric. Finally, we investigate the completeness property of the spaces of -bounded and -continuous functions.

Let be compact. Then, by , we denote the space of -continuous functions defined on the set . One can easily see that the set forms a vector space over with respect to the algebraic operations addition and scalar multiplication defined on as follows:

In order to show that is a metric space with the metric defined by (32), we give the following lemma.

Lemma 9. is a metric space.

Proof. Let and be the generators on the sets of arguments and values, respectively. (i)For every and for every , we have
that is, (M1) holds.(ii) One can easily see for every that
which shows that the symmetry axiom (M2) also holds.(iii)By a routine verification for every , if we apply to then we obtain that
which yields by applying that
Therefore, by taking maximum, one can derive that
This means that the triangle inequality (M3) also holds.Therefore, since (i)–(iii) are satisfied, is a metric on . This completes the proof.

Definition 10. A -norm is a nonnegative -real valued function on whose value at an is denoted by , that is, , and satisfies the following conditions: (N1),(N2) (absolute homogeneity),(N3) (triangle inequality),for all and for all scalars .

The -norm on defines a metric on given by
and is called the induced -metric by the -norm.

The definition of space of continuous functions makes it possible to give a much more intuitive meaning to the classical notion of uniform convergence. Convergence in the space of continuous functions space turns into the uniform convergence. One of the most important results of the concept of the space of continuous functions is the famous Stone-Weierstrass approximation theorem which is a very powerful tool for proof of general results on continuous functions. Using this theorem, we can prove some results fits for functions of special type and later extend them to all continuous functions by a density argument. In this paper, we show, with the rules of non-Newtonian calculus, its advantages to Stone-Weierstrass theorem in the space of , or not. The answer of this question is affirmative in some cases, but not every time when we want. In [3], Uzer showed by using multiplicative calculus which is a kind of non-Newtonian calculus that it is more flexible than the classical calculus for Bessel functions in a special domain. We can reproduce more examples for the same situation but we mainly focused on the theoretical properties of the space .

Theorem 11. is a normed space with the norm given by

Proof. Let and . Then, the following hold.(i)One can easily show that
That is to say that the axiom (N1) holds.(ii)From the property of vector space axioms of the space , it is immediate that
Hence, the absolute homogeneity axiom (N2) also holds.(iii)It is obtained by the similar way used in the proof of Lemma 9 that
This means that the triangle inequality axiom (N3) is satisfied.Since (i)–(iii) are fulfilled, , defined by (41), is a norm for the space .

Definition 12. Let be any set and let be a complex normed space. A mapping from into is bounded if is bounded in , or equivalently if is finite. The set of all bounded mappings from into is denoted by .

Corollary 13. The set of all bounded mappings from into is denoted by is a complex vector space, since
Moreover, on this space,
is a norm, as can be easily verified.

Theorem 14. is a Banach space if is a Banach space.

Proof. Let be a Cauchy sequence in . Then, for any , there is an such that for all . From norm, it follows for any that we have for . Hence, the sequence converges to an element , since is complete. Furthermore, we have for any and . By the -triangle inequality given by Lemma 4, we first deduce that for all ; hence is bounded. Moreover, we have for all and this means that the sequence converges to in the space .

It is known from Mathematical Analysis that when we applied to Non-Newtonian calculus if is a sequence of functions from into a metric space , we say the sequence is -convergent to a function from into if, for each , the sequence -converges in to ; we call that -converges uniformly on to if the following equality holds:
It is obvious that -uniform convergence implies -simple convergence; however, the converse is not true. If is a normed space, then -convergence of a sequence of functions in , therefore, corresponds to -uniform convergence of the sequence in .

Finally, we give the theorem on the completeness of the space of -continuous functions.

Proof. Let be any Cauchy sequence in . For each , we have
and so is a Cauchy sequence of -real numbers, and is a Cauchy sequence of real numbers as well; hence they are -convergent and convergent, respectively. Let be defined by or for each . Since, given any ,
by letting , we obtain independent of that
for sufficiently large . Hence, uniformly in .Finally, since the limit of a uniformly convergent sequence of continuous functions is continuous, then and as . This completes the proof.

4. Applications

In this section, we study some properties of multiplicative calculus which is a kind of (NC). This calculus may be created by taking , the identity function, and , the exponential function. Multiplicative calculus, in short (MC), has many applications in some branch of mathematics such as financial mathematics and elasticity. In the present paper, we investigate the complex multiplicative functions in a complex domain to make rational approximation for analytic functions. Later, as an application, we mention the inner product property of (MC).

Let be a single-valued function defined on a set which is dense itself; that is, every point of is a limit point of . Then, is said to be locally analytic on if, given any , there is a neighborhood and a power series such that for all . The concept of a locally analytic function on the domain reduces to a single-valued analytic function.

In (NC), particularly in (MC), when we consider single-valued analytic functions on a domain, we consider the product, , where is the th coefficient of the Taylor product of the function at such that is the th order -derivative of the function .

This is an application to show rational approximation also applicable for -calculus (Multiplicative calculus). One question arises that which way of approximation is better, the classical one or the new one? Uzer made some numerical solutions for the Bessel differential equations in [3] and he considered the function with the modulus . It is suggested that if the solution function varies exponentially along a specific contour, then the method in the (MC) sense shows a better performance. Otherwise, that is if the solution function is oscillatory or linearly varying, the method in the (CC) sense will be better. The solution in the given example exhibits exponential variations everywhere on the complex plane except near the real axis. The other elements of the family of Bessel functions also exhibit exponential variations. Indeed, there are many other functions exhibiting exponential variations on the complex plane such as the famous sigmoid function which plays an important role in decision making in Neutral Networks.

As a second application of (MC), we mention the -inner product property. A -inner product space is a vector space with an inner product defined on it. A -norm is defined by and if holds, then and are called -orthogonal vectors. A -Hilbert space is a complete -inner product space. The spaces to be considered are defined as follows.

Definition 16. Let be a vector space over the field or . A -inner product on is a mapping from into the scalar field (or ) of ; that is, with every pair of vectors and , there is a scalar called the -inner product of and , such that for all vectors and any scalar , the following axioms hold: (IP1) ,(IP2) ,(IP3) ,(IP4) and . Then, we say that is an inner product space provided (IP1)–(IP4) hold. Here, denotes the complex conjugate of . The conjugate of a -complex number is . Note that (IP2) and (IP3) imply that .

Corollary 17. It can easily be seen from the equality (24) and from [10] that
where and , . For example, if and , then one has

Remark 18. Since the product of with itself equals , we may define to be . Of course, the product of with itself also equals . Therefore, in (MC), turns out .

If , then, using again the equalities (24) and [10, p. 88], we conclude that
Thus, in (MC), we have .

In a (real or complex) -inner product space , two vectors are called orthogonal and we write provided . For a subset , the set is defined by

Corollary 19. A multiplicative inner product space satisfies the parallelogram equality. Let and such that . Consider
Of course, if we take and , we can easily obtain these results for (MC).

Definition 20. Let be a -normed space. If the corresponding metric is complete, we say that is a Banach space. If is an -inner product space whose corresponding metric is complete, we say that is a -Hilbert space.

Theorem 21 (Cauchy-Schwartz inequality). For all , the following inequality holds:
where , , , and .

Proof. Let . Then,
If we apply the function of to (57), then we have
which gives by applying that
which completes the proof.

Corollary 22. The space is an -inner product space but is not a Hilbert space with the integral metric defined by
where .

The proof is easily obtained by the appropriate verifications. Indeed, if we take , we obtain the classical calculus (CC) and, in (CC), the results are the same for Corollary 22. It is an expected situation, because (CC) is a kind of (NC) and we cannot generalize the assertion of Corollary 22 differently.

Let us consider the space with the inner product
which gives the associated norm . The inner product space is not complete; the space of Riemannian integrable functions on the interval that are square-integrable, that is,
is not complete.

As a final application, we give an inclusion relation between the spaces and , the space of first-order -differentiable functions in .

Theorem 23. and the inclusion is strict.

Proof. In [10], if is a -continuous function in a given point , then from the definition we have if and only if is -convergent to whenever any sequence